Research map / Learning theory and algorithms / Matrix and tensor methods
Weakly Convex Regularisers for Inverse Problems: Convergence of Critical Points and Primal-Dual Optimisation
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Shows that weakly convex regularisers give convergence of critical points for variational inverse problems, proves convergence and an O(log k / k) ergodic rate for PDHGM under weak convexity and KL, proves universal approximation for input weakly convex networks, and tests adversarial weakly convex regularisers on CT reconstruction.
Problem
Convergent regularisation for inverse problems has been studied mainly for global minimisers, but practical optimisation of learned non-convex regularisers only reaches critical points; convergence of critical points and provable primal-dual optimisation for weakly convex regularisers were lacking.
Approach
Impose weak convexity on the regulariser (via input weakly convex neural networks) to get existence, stability and convergence of critical points, prove convergence of PDHGM (theta=1) for weakly convex regularisers with KL-based ergodic rate, and define an adversarial weakly convex regulariser (AWCR) learned in the weakly supervised setting.
Key results
- On sparse-view CT, AWCR-PD reaches 36.9993 dB PSNR and 0.9108 SSIM, versus 36.4079 dB for AR, approaching the supervised U-Net post-processing at 37.1075 dB.
- On limited-angle CT, AWCR attains 26.7233 dB PSNR, slightly above ACNCR (26.5420 dB) and ACR (26.4459 dB), but lower SSIM than ACR.
- Theory: PDHGM with theta=1 on weakly convex regularisers has an O(log k / k) ergodic primal-dual gap under a Kurdyka-Lojasiewicz condition.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use AWCR instead of AR | Weakly supervised CT reconstruction · Sparse-view CT | PSNR (dB): 36.4079 -> 36.5323 | Helped |
| Use AWCR instead of AR | Weakly supervised CT reconstruction · Limited-view CT | PSNR (dB): 23.6475 -> 26.7233 | Helped |
| Use AWCR instead of AR | Weakly supervised CT reconstruction · Sparse-view CT | SSIM: 0.9101 -> 0.9046 | Hurt |
| Use AWCR instead of AR | Weakly supervised CT reconstruction · Limited-view CT | SSIM: 0.6257 -> 0.8105 | Helped |
| Use AWCR instead of ACR | Weakly supervised CT reconstruction · Limited-view CT | PSNR (dB): 26.4459 -> 26.7233 | Helped |
| Use AWCR instead of ACR | Weakly supervised CT reconstruction · Limited-view CT | SSIM: 0.8184 -> 0.8105 | Hurt |
| Use PDHGM to solve AWCR instead of the subgradient method | AWCR CT reconstruction · Sparse-view CT | PSNR (dB): 36.5323 -> 36.9993 | Helped |
| Use PDHGM to solve AWCR instead of the subgradient method | AWCR CT reconstruction · Limited-view CT | PSNR (dB): 26.7233 -> 24.3254 | Hurt |
| Use PDHGM to solve AWCR instead of the subgradient method | AWCR CT reconstruction · Limited-view CT | SSIM: 0.8105 -> 0.7650 | Hurt |
Surprising findings
- AWCR-PD performs worse than AWCR in limited-angle CT (24.33 dB vs 26.72 dB PSNR), attributed to severe ill-posedness of forward and adjoint operators.
- A weakly convex regulariser with bounded, coercive form can still have critical points that do not converge in infinite dimensions, giving a counterexample to weak-convergence results without extra assumptions.
Limitations
- Convergence of PDHGM requires iterate boundedness as an assumption.
- Analysis is restricted to finite-dimensional Hilbert spaces for strong convergence; infinite-dimensional case needs additional assumptions.
- Performance in limited-angle CT of the primal-dual variant is lower than the non-primal-dual AWCR.
Experimental setup
Benchmarks: CT reconstruction sparse-view CT reconstruction limited-angle
Metrics: PSNR SSIM
Baselines: FBP TV LPD U-Net AR ACR ACNCR
Builds on: adversarial regularisation input convex neural network PDHGM Wasserstein GAN learned primal-dual
Closest papers by meaning
- What's in a Prior? Learned Proximal Networks for Inverse Problems (ICLR 2024 · similarity 0.87)
- Controlling the Flow: Stability and Convergence for Stochastic Gradient Descent with Decaying Regularization (NeurIPS 2025 · similarity 0.87)
- QN-Mixer: A Quasi-Newton MLP-Mixer Model for Sparse-View CT Reconstruction (CVPR 2024 · similarity 0.86)
- Cauchy-Schwarz Regularizers (ICLR 2025 · similarity 0.86)
- Iterative Regularization with k-support Norm: An Important Complement to Sparse Recovery (AAAI 2024 · similarity 0.86)
- A Restoration Network as an Implicit Prior (ICLR 2024 · similarity 0.85)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.